Abstract
The aim of this paper is to point out recent and classical results related with the existence of solutions of second-order problems coupled with nonlinear boundary value conditions.
Highlights
The first steps in the theory of lower and upper solutions have been given by Picard in 1890 1 for Partial Differential Equations and, three years after, in 2 for Ordinary Differential Equations
There have been a large number of works in which the method has been developed for different kinds of boundary value problems, first, second- and higher-order ordinary differential equations with different type of boundary conditions such as, among others, periodic, mixed, Dirichlet, or Neumann conditions, have been considered
Lepin et al generalize in 31, 34, 35 some of the results proved by Erbe in 41
Summary
The first steps in the theory of lower and upper solutions have been given by Picard in 1890 1 for Partial Differential Equations and, three years after, in 2 for Ordinary Differential Equations. The problem of finding a solution of the considered problem is replaced by that of finding two well-ordered functions that satisfy some suitable inequalities Following these pioneering results, there have been a large number of works in which the method has been developed for different kinds of boundary value problems, first-, second- and higher-order ordinary differential equations with different type of boundary conditions such as, among others, periodic, mixed, Dirichlet, or Neumann conditions, have been considered. In 1967, Kiguradze 18 proved that it is enough to consider a one-sided Nagumo condition by eliminating the absolute value in 1.11 to deduce existence results for Dirichlet problems. Cherpion et al prove in 37 the existence of extremal solutions for the Dirichlet problem without assuming the condition 1.20 They consider a more general problem: the φ-laplacian equation. Similar results are deduced for the periodic boundary conditions in 12 In this case the arguments follow from the finite intersection property of the set of solutions see 38, 39
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